If it were in the passé composé, both tout le/un mois would be possible depending on what you want to say.

Son père a été malade tout le/un mois.

Tout le mois would mean "the entire month," hence referring to a specific month known from context (may be a past, present, or future month). On the other hand, tout un mois would mean "an entire month," hence referring to the duration itself, without referring to any month in particular.